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WHEN MATHEMATICS TEACHERS CONSIDER ACTING ON BEHALF OF THE

DISCIPLINE, WHAT ASSUMPTIONS DO THEY MAKE?

!!!Amanda!Milewski! ! !!Ander!Erickson!!!!!!!!!!!!!!!!!Patricio!Herbst!!!!!!!!!!!!!!!!!Justin!Dimmel!

University!of!Michigan!!!!!!!!!University!of!Michigan!!!!!!!University!of!Michigan!!!!University!of!Maine!

!

We discuss findings from a study of 226 K-12 mathematics teachers who rated the appropriateness

of instructional actions in a scenario-based assessment. The teaching actions shown in the

scenarios were designed to represent mathematics teachers fulfilling their obligation to the

discipline of mathematics. Our analysis of the participants’ follow-up responses provides insight

into how other professional obligations interact with teachers’ obligation to the discipline. We

describe our method for detecting what we call conditional construals—assumptions participants

regarded as critical for rating the appropriateness of instructional actions. In addition, we report

preliminary findings about how those assumptions can be accounted for in terms of the professional

obligations of mathematics teaching.

Keywords: Instructional Activities and Practices, Teacher Knowledge, Teacher Beliefs, Research

Methods

Introduction

Scenario-based assessments are widely used to assess judgment in professional fields, such

as medicine, police work, army tactics, and human resources (Weekley & Polyhardt, 2006). In this

study, we analyzed a set of open-ended responses to scenario-based assessment items of K-12

mathematics teaching. Open responses from participating teachers followed forced-choice ratings

of instructional actions shown within scenarios of K-12 mathematics classrooms. There were two

parts to the analysis. The first part of the analysis involved validating a procedure for identifying

when a response contained linguistic markers that indicated that the appropriateness of an

instructional action was dependent on an element of the scenario (e.g., traits of the students, the

amount of available time, the content that had been previously covered) that was unspecified in the

representation that they viewed. We call such markers conditional construals. For the first part of

the analysis, we analyzed responses from 226 teachers to 15 items (n=3405) designed to probe

teachers’ professional obligation to the discipline of mathematics (Herbst & Chazan, 2012). The

second part of the analysis was a qualitative examination of the kinds of additional information

participants stated they would need to know about a scenario in order to rate the appropriateness of

an instructional action. The qualitative analysis examined the responses to 2 of the 15 items that had

the highest incidence of responses that contained a conditional construal. In this paper we report on

the results of the preliminary qualitative analysis of the responses to these 2 high construal items.

We looked for additional aspects of the represented scenario that teachers considered critical

in deciding whether an action was appropriate. The following research questions guided our

inquiry: 1) What types of assumptions do teachers make in order to justify instructional decisions

that respond to a professional obligation to the discipline? 2) How do the different professional

obligations support or constrain teachers’ departures from “business as usual” in favor of their

obligation to the discipline of mathematics? Our purpose with this study is to investigate the types

of assumptions teachers make when considering instructional actions that respond to the

professional obligation to the discipline of mathematics.

Theoretical Framework for Research

Herbst and Chazan’s (2012) account of the practical rationality of mathematics teaching

attempts to synthesize rational actor theories of teacher decision making (e.g., Calderhead, 1996;

Schoenfeld, 2010) with accounts of teaching as a sociocultural activity (Stigler & Hiebert, 1999).

Herbst and Chazan (2012) propose that while teachers have personal resources such as knowledge

and beliefs, teachers also play roles in activity systems (e.g., algebra instruction) that have

customary norms (Much & Shweder, 1978). Within such activity systems, teachers are accountable

to four professional obligations—to the discipline of mathematics, to individual students, to the

classroom community, and to the institutions of schooling (Herbst & Chazan, 2012). While actions

following customary norms may be enacted without reflection, a teacher’s deviations from those

norms, whether motivated by individual resources or in response to social demands, need

justification (Buchmann, 1986). Herbst and Chazan (2012) argue that the four obligations provide a

basis for the sources of professional justification. For example, while the obligation to the discipline

of mathematics would require a teacher to limit the range of symbolic strings that she allows herself

to call equations (e.g., 2x = 5 is an equation but 3x -7 is not an equation), this same obligation could

also justify her calling equation a symbolic string like 2x = x2 in spite of the fact that the beginner’s

algebra curriculum might not include it under the equations that algebra students learn to solve

(Herbst & Chazan, 2012).

While much work has been carried out to establish links between teachers’ beliefs and

conceptions about mathematics and instructional practices (Skott, 2001; Stipek et al., 2001), the

notion of practical rationality suggests that teachers’ decision-making is not driven by individual

resources alone. Rather, teachers’ decisions are context-dependent and a fuller understanding of the

resources that they can employ for departing from normative practice can be achieved through the

use of instruments that represent the situated nature of practice. As such, we approach the work of

understanding teachers’ decision making by presenting them with scenarios of instruction in which

the represented teacher departs from instructional norms in order to act on behalf of the professional

obligation to the discipline.

Methodology

We conducted our study on data collected from K-12 inservice mathematics teachers with a

15 item scenario-based instrument designed to measure recognition of teachers’ professional

obligation to the discipline of mathematics. Within each item, participants examined a scenario in

which a teacher departs from an instructional norm in order to attend to one of the professional

obligations. Participants were asked to rate the extent of their agreement (using a 6-point Likert-like

response format from strongly disagree to strongly agree) with the teacher’s action. Participants

were then prompted to comment on their ratings. In order to identify conditional construals in these

responses, we located circumstances of contingency (Halliday & Matthiessen, 2004, p. 271) which

are accompanied by linguistic markers such as “depending on”, “as long as”, and “assuming”. In

particular, we examined the nominal group introduced by the circumstances of contingency. These

nominal groups are usually “a noun denoting an entity whose existence is conditional, a noun

denoting an event that might eventuate, or a nominalization denoting a reified process or quality”

(Halliday & Matthiessen, 2004, p. 272). We present an example with the following response, “This

depends on timing. I think it can be valuable information for students to see common errors played

out in a solution, but time will dictate whether or not that can happen in any given scenario.” We

argue that “timing” is marked here as a circumstance of contingency by the prepositional phrase

“this depends on”. We then categorized those circumstances by professional obligation. In this

example, we placed the conditional construal in the institutional category because the participant

indicates that the appropriateness of the action is contingent on the amount of time available, and

time constraints are imposed by the institution.

Findings

We found that teachers consider the other obligations—to the students as individuals, to the

classroom community, and to the institution—as they consider whether to respond to their

professional obligation to the discipline of mathematics. We illustrate these categories of

conditional construals by sharing teachers’ responses from the two items with the highest number of

responses that contained conditional construals. The analysis reported below qualitatively assesses

the content of these conditional construals, with respect to the professional obligations.

In the first item, participants view a scenario in which the teacher announces that students

seem to be on the right track and, rather than spend time on practice problems (which we

hypothesize would be the normative action in such a situation), the teacher decides to discuss

mathematical theory related to the topic at hand. Participants were asked to rate the statement: “The

teacher should give students additional practice problems, rather than elaborate on mathematical

theory.” Of the 226 participant responses, 16% indicated that their decision depended on

assumptions they had made about the scenario. In the second item, participants viewed a scenario in

which a student makes a mistake while working a problem at the board; rather than point out the

mistake directly (which we hypothesize would be the normative action in such a situation), the

teacher allows the student to build on the mistake. Such an action could be justifiable on the

disciplinary grounds that the teacher is creating an opportunity for a contradiction or anomaly to

arise from mathematical reasoning that builds on the mistake. Participants were asked to rate the

statement: “The teacher should correct the mistake observed, rather than ask students to build on

mistaken work,” and we found that 7% of the participants indicated that their decision depended on

assumptions they made about the scenario.

These items highlight the variety of factors that teachers consider in order to make

instructional decisions that respond to the disciplinary obligation and demonstrate how the role of

the other professional obligations can vary depending on the scenario. For example, while there

were 30 participants who referenced the individual obligation in their responses to the first item, the

individual obligation was used in different ways. One perspective suggested that knowing the

“level” of the class is essential for determining whether or not a particular instructional action is

appropriate; another perspective suggested that it was of primary importance to determine whether

students had been taught relevant information. While these are both manifestations of the obligation

to the individual student, they have different implications for how the teacher perceives the agency

of the students and the responsibility of the teacher. Such differences manifested themselves across

items as well: In the case of the first item, there were 30 references to the individual obligation, but

there were only 9 references to the individual obligation for the second item. Ongoing analysis is

being to conducted to perceive patterns in the way each of the obligations are manifested within

participants' construals.

Study’s Significance

Mathematics instruction is beholden to the discipline of mathematics as the source of

legitimacy of the knowledge at stake in the classroom as well as the practices that govern the

development of that knowledge. Nonetheless, there is often a tension between the type of

instruction advocated by mathematicians and mathematics educators and the everyday reality of K-

12 education (Schoenfeld, 2004). We have demonstrated how we have been able to collect a corpus

of data from K-12 teachers of mathematics describing the circumstances that dictate whether they

approve of a fellow teacher departing from customary classroom practice in order to respond to

their perceived professional obligation to the discipline of mathematics. Our analysis of that corpus

suggests that, for these teachers, their professional obligation to the individual students (e.g.

students’ understanding), to the discipline of mathematics (e.g. mathematical value), to the

classroom community (e.g. class norms), and to the institution of school (e.g. time) all have a

bearing on their approval of such decisions. It remains an open question for us whether individual

differences account for part of the variance in participants’ conditional construals. To explore this,

we could examine how proxies for such differences, such as mathematical knowledge for teaching

and experience, are related to the ways that they see the professional obligations impinging on their

practice.

While surveys of teachers’ beliefs about mathematics provide one way of conceptualizing

how the discipline of mathematics may influence instructional decisions, our work presents a

different approach by suggesting that there are aspects of the professional position (for example, the

need to teach a given curriculum, a limited allowance of time, the characteristics of the class being

taught, the various characteristics and needs of individuals one is assigned to teach) that may

mediate a teachers’ willingness to respond to the discipline regardless of their beliefs about the

subject. As a better model is developed of the way that the professional obligations interact both

with one another and with personal resources of the mathematics teacher, and as the different

aspects of the professional obligations themselves are further analyzed as we have begun to do here,

we will be closer to being able to understand the hidden factors that obstruct or facilitate the take-up

of instructional practices that depart from the norm in order to align with the discipline of

mathematics.

References

Buchmann, M. (1986). Role over person: Morality and authenticity in teaching. The Teachers College Record, 87(4),

529-543.

Calderhead, J. (1996). Teachers: Beliefs and knowledge. In D.C Berliner & R.C. Calfee (Eds.), Handbook of

educational psychology., (pp. 709-725). New York: Macmillan.

Halliday, M., Matthiessen, C. M., & Matthiessen, C. (2014). An introduction to functional grammar. Routledge.

Herbst, P., & Chazan, D. (2012). On the instructional triangle and sources of justification for actions in mathematics

teaching. ZDM, 44(5), 601-612.

Much, N. C., & Shweder, R. A. (1978). Speaking of rules: The analysis of culture in breach. New Directions for Child

and Adolescent Development,1978(2), 19-39.

Schoenfeld, A. H. (2010). How we think: A Theory of Goal-Oriented Decision Making and its Educational

Applications. New York: Routledge

Schoenfeld, A. H. (2004). The math wars. Educational policy, 18(1), 253-286.

Skott, J. (2001). The Emerging Pracitces of a Novice Teacher: The Roles of His School Mathematics Images. Journal

of Mathematics Teacher Education, 4(1), 3-28.

Stigler, J. W., & Hiebert, J. (2009). The teaching gap: Best ideas from the world's teachers for improving education in

the classroom. Simon and Schuster.

Stipek, D. J., Givvin, K. B., Salmon, J. M., & MacGyvers, V. L. (2001). Teachers’ beliefs and practices related to

mathematics instruction. Teaching and teacher education, 17(2), 213-226.

Weekley, J. A., & Ployhart, R. E. (Eds.). (2013). Situational judgment tests: Theory, measurement, and application.

Psychology Press.